The polynomialhierarchy is a hierarchy of complexity classes generalizing the relationship between

The polynomialhierarchy is closely related to the arithmetical hierarchy; indeed, an alternate definition is almost identical to the definition of the arithmetical hierarchy but stricter rules on what quantifiers can be used.

This is version 3 of polynomialhierarchy, born on 2002-09-06, modified 2003-12-02.

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene hierarchy classifies the set of arithmetic formulas (or arithmetic sets) according to their degree of solvability.

Layers in the hierarchy are defined as those formulas which satisfy a proposition (description) of a certain complexity.

The polynomialhierarchy is a "feasible resource-bounded" version of the arithmetical hierarchy, in which polynomial length bounds are placed on the strings involved, or equivalently, polynomial time bounds are placed on the Turing machines involved.

The classes (A registered nurse who has received special training and can perform many of the duties of a physician) NP and (Click link for more info and facts about co-NP) co-NP can be defined as, and, where (The 16th letter of the Roman alphabet) P is the class of all feasibly (polynomial-time) decidable languages.

The polynomialhierarchy is an analogue (at much lower complexity) of the (Click link for more info and facts about exponential hierarchy) exponential hierarchy and (Click link for more info and facts about arithmetical hierarchy) arithmetical hierarchy.

The complexity class AM (or AM[1]) is the set of decision problems that can be decided in polynomial time by an Arthur-Merlin protocol where Arthur communicates first, Merlin replies and both of them can only send one message to the other party.

Both MA and AM are contained in the polynomialhierarchy.

MA is also contained in NP/poly, the class of decision problems computable with in non-deterministic polynomial time with a polynomial size advice.

In Pollett~\cite{cpollett00} a bounded arithmetic theory $Z$ was shown not to be able to prove the collapse of the polynomialhierarchy.

Namely, it can "reason" about all the functions in the log-time hierarchy, it can prove the log-time hierarchy differs from NP, and, in fact, we give some evidence it might be able to show the log-time hierarchy is infinite.

We then use these results together with recent results about bounded query classes to derive tighter collapses of the polynomialhierarchy under the assumption that various bounded arithmetic theories are equal.

www.cs.sjsu.edu /faculty/pollett/papers (2908 words)

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A functional of type two is a polynomial-time computable if it is computed by a deterministic function-oracle Turing machien whose runtime is bounded by a polynomial that does not depend on the choice of oracle functions.

Townsend also introduced a boldface polynomialhierarchy of type two by a relativization method, and gave a ``topological'' characterization of the first level of this hierarchy.

We reformulate Townsend's topological notions associated with time bounded computations of function-oracle Turing machines, and further extend his ``topological'' characterization to all levels of the boldface polynomialhierarchy of type two.

What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses - ...(Site not responding. Last check: 2007-11-06)

The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels.

Bounded queries to SAT and the Boolean hierarchy(Site not responding. Last check: 2007-11-06)

We also consider the similarly-defined hierarchies of functions that can be computed by a polynomial-time Turing machine that makes a bounded number of queries to an NPoracle.

In addition we show that the Booleanhierarchy and the bounded query hierarchies (of languages) either stand or collapse together.

Finally we show that if the Booleanhierarchy collapses to any level but the zeroth (deterministic polynomial time), then for all k there are functions computable in polynomial time with k parallel queries to an NP set that are not computable in polynomial time with

The higher a formalism is in the model hierarchy, the more its efficiency in representing models is -- and analogously for theorems.

In Theorems 5 and 7 we assume that the polynomialhierarchy does not collapse.

For example, as a consequence of Theorems 3 and 7 there is no poly-size reduction from PL to the syntactic restriction of PL allowing only Horn clauses that preserves the theorems, unless the polynomialhierarchy collapses.

Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given.

Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g_i is exactly m.

As a consequence, we obtain new complexity bounds for polynomial system solving which are (a) expressible in terms of Newton polytope volumes, and (b) completely general, free of non-degeneracy assumptions.

The scheme for the multivariate polynomial is an example of a multi-oracle instance-hiding scheme.

Interestingly, this low-degree polynomial trick, which was devised in order to construct instance-hiding schemes, became a crucial ingredient in the characterization of the set-recognition power of interactive proof systems, both one-prover and multiprover.

The zero'th level of the hierarchy is polynomial time, and the first is nondeterministic polynomial time.

In other words a problem X is if and only if there exists some problem Y in such that X is polynomial-time Turing reducible to Y. This means that given oracle for Y there exists an algorithm solves X in polynomial time (possibly by using that oracle).

There are algorithms as Quicksort that can sort the list using a polynomial number of calls to the routine plus a polynomial amount of additional Therefore sorting is NP-easy.

We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet.

The testers in the latter have the goal of determining whether a program computes a polynomial of given degree, whereas we are interested in checking the properties of a given polynomial.

All current polynomial time methods (to our knowledge) are unlikely to recover the topology of the true tree from sequences of realistic lengths (bounded, perhaps, by 10,000 nucleotides) for large sets of widely divergent sequences, even if such methods are known to be able to reconstruct the correct topology given long enough sequences.